Question

The diameter and height of a cylinder are equivalent to the edge lengths of a cube. Which figure has a greater volume?


They have the same volume.



The cube



The cylinder



There is not enough information.

Answer 1

Let's denote the edge length of the cube as \( s \).

For the cube:
- The volume \( V_{\text{cube}} \) is given by the formula:
\[
V_{\text{cube}} = s^3
\]

For the cylinder:
- The diameter is equal to the edge length of the cube, so the diameter \( d = s \).
- The radius \( r \) of the cylinder is half the diameter:
\[
r = \frac{s}{2}
\]
- The height \( h \) of the cylinder is also equal to the edge length of the cube, \( h = s \).
- The volume \( V_{\text{cylinder}} \) is given by the formula:
\[
V_{\text{cylinder}} = \pi r^2 h = \pi \left(\frac{s}{2}\right)^2 s = \pi \left(\frac{s^2}{4}\right) s = \frac{\pi s^3}{4}
\]

Now we compare the volumes:
- Volume of the cube: \( V_{\text{cube}} = s^3 \)
- Volume of the cylinder: \( V_{\text{cylinder}} = \frac{\pi s^3}{4} \)

To compare \( s^3 \) and \( \frac{\pi s^3}{4} \):
- We can factor out \( s^3 \) (assuming \( s > 0 \)):
\[
s^3 \quad \text{and} \quad \frac{\pi}{4} s^3
\]
- Since \( \frac{\pi}{4} \approx 0.785 \), we know \( s^3 > \frac{\pi}{4} s^3 \).

Thus, the volume of the cube is greater than the volume of the cylinder.

Therefore, the answer is:
**The cube**.